Optimal. Leaf size=137 \[ \frac{a b \left (19 a^2+16 b^2\right ) \sinh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}+\frac{7 a b \sinh (c+d x) (a+b \cosh (c+d x))^2}{12 d} \]
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Rubi [A] time = 0.150763, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2656, 2753, 2734} \[ \frac{a b \left (19 a^2+16 b^2\right ) \sinh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}+\frac{7 a b \sinh (c+d x) (a+b \cosh (c+d x))^2}{12 d} \]
Antiderivative was successfully verified.
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Rule 2656
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x))^4 \, dx &=\frac{b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d}+\frac{1}{4} \int (a+b \cosh (c+d x))^2 \left (4 a^2+3 b^2+7 a b \cosh (c+d x)\right ) \, dx\\ &=\frac{7 a b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{12 d}+\frac{b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d}+\frac{1}{12} \int (a+b \cosh (c+d x)) \left (a \left (12 a^2+23 b^2\right )+b \left (26 a^2+9 b^2\right ) \cosh (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a^4+24 a^2 b^2+3 b^4\right ) x+\frac{a b \left (19 a^2+16 b^2\right ) \sinh (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{24 d}+\frac{7 a b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{12 d}+\frac{b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.211921, size = 104, normalized size = 0.76 \[ \frac{12 \left (24 a^2 b^2+8 a^4+3 b^4\right ) (c+d x)+24 b^2 \left (6 a^2+b^2\right ) \sinh (2 (c+d x))+96 a b \left (4 a^2+3 b^2\right ) \sinh (c+d x)+32 a b^3 \sinh (3 (c+d x))+3 b^4 \sinh (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 119, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{4} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +4\,a{b}^{3} \left ( 2/3+1/3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \right ) \sinh \left ( dx+c \right ) +6\,{a}^{2}{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,{a}^{3}b\sinh \left ( dx+c \right ) +{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04603, size = 247, normalized size = 1.8 \begin{align*} \frac{1}{64} \, b^{4}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{3}{4} \, a^{2} b^{2}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{4} x + \frac{1}{6} \, a b^{3}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} - \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{4 \, a^{3} b \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12324, size = 296, normalized size = 2.16 \begin{align*} \frac{{\left (3 \, b^{4} \cosh \left (d x + c\right ) + 8 \, a b^{3}\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 3 \,{\left (b^{4} \cosh \left (d x + c\right )^{3} + 8 \, a b^{3} \cosh \left (d x + c\right )^{2} + 32 \, a^{3} b + 24 \, a b^{3} + 4 \,{\left (6 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.40657, size = 240, normalized size = 1.75 \begin{align*} \begin{cases} a^{4} x + \frac{4 a^{3} b \sinh{\left (c + d x \right )}}{d} - 3 a^{2} b^{2} x \sinh ^{2}{\left (c + d x \right )} + 3 a^{2} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac{3 a^{2} b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 a b^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 a b^{3} \sinh{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac{3 b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{5 b^{4} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \cosh{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12988, size = 259, normalized size = 1.89 \begin{align*} \frac{3 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 144 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 384 \, a^{3} b e^{\left (d x + c\right )} + 288 \, a b^{3} e^{\left (d x + c\right )} + 24 \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )}{\left (d x + c\right )} -{\left (32 \, a b^{3} e^{\left (d x + c\right )} + 3 \, b^{4} + 96 \,{\left (4 \, a^{3} b + 3 \, a b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} + 24 \,{\left (6 \, a^{2} b^{2} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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